On 3/23/2013 6:46 PM, Ross A. Finlayson wrote: > On Mar 23, 3:34 pm, fom <fomJ...@nyms.net> wrote: >> On 3/23/2013 5:09 PM, Ross A. Finlayson wrote: >> >> >> >> >> >> >> >> >> >>> On Mar 23, 2:44 pm, fom <fomJ...@nyms.net> wrote: >>>> On 3/23/2013 4:34 PM, Ross A. Finlayson wrote: >> >>>>> In a sense, infinity _is_ the numbers. Start from even more >>>>> fundamental objects than natural numbers as elements. Like the >>>>> numbers, they are as different as they can be and as same as they can >>>>> be, where they are each different in not being any other and each same >>>>> in being defined by that difference. There's no stop to that, it's >>>>> gone on, forever. Then, in a way like when you look into the void, it >>>>> looks into you >> >>>> Is this your way of saying that if you look >>>> into the void, you and the void become one? >> >>>> http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_theorem >> >>>> Just kiddding.... >> >>>> I think Cantor would appreciate your sentiment that >>>> the numbers of Cantor's paradise are more fundamental >>>> than those of Kronecker's torment. >> >>> I wouldn't say that infinity, even in the numbers, is either of those >>> things. In ZF, Infinity is _axiomatized_ to be an inductive set, and >>> a well-founded/regular one, that's not a given. Calling that the >>> universe, Russell's comment is that it would contain itself. >> >>> There's a case for induction, as it were, that each case exists. Then >>> it is to be of deduction, not fiat by axiomatization, from simple >>> principles of constancy and variety, the continuum. >> >>> In a theory with sets as primary objects, a set theory and a pure set >>> theory, numbers would be very rich objects indeed, as not just >>> individual elements by their elements, but all relations of numbers. >>> Set theory (well-founded, as it were, regular or that objects are >>> transitively closed) is at once over-simplification, to talk about >>> anything besides sets, and over-complexification, to talk about itself >>> when any universal statement is in the meta. >> >>> There are no numbers in a pure set theory. To call the natural >>> integers a set, it contains only numbers, for the Platonists: elements >>> of the structure, of numbers, as: none exist in a void. >> >> It is odd. In some sense, modern mathematics actually >> treats its objects as urelements relative to set theory. >> Looking at Hilbert, he makes statements whereby his formalism >> is intended to supersede the class-based constructions of >> Dedekind. >> >> Your frank statement that a set is not a number reflects >> that sentiment. > > Particular finite sets are called ordinals, set-theoretic operations > on them are defined that give the same results as Presburger/Peano > arithmetic of the natural integers. The negative integers aren't > simply the complement as in finite-word-width machine arithmetic, but > again simple enough operations on sets (with the only ur-element being > the empty set) give a "model" of the integers. Rationals are defined > simply enough as equivalence classes over any pairs of integers, > besides zeros, the reals then see the Least Upper Bound as axiom. > These are all to match number-theoretic features, and largely suffice > for integers and rational numbers, but not so obviously do sets > suffice to represent thusly elements (and all of) the continuum of > real numbers. > > Then, though, to call the empty set the number zero: wouldn't that be > the number zero wherever there's an empty set? Building upwards to > have particular sets for each of of the finite integers: then to > build the numbers as sets, is to build all the relations of the > numbers as sets, not just as to a set-theoretic model of only that set > of numbers' operations: but of all instances, besides the schema. > Where the ur-element is any thing, it so implies all other things, > and is so implied. The collection and aggregates of sets or > categorization or refinement of types or partition or bounding of > division, are all of the same corpus. > > Here back to the questions as above: > > 1) is not forcing simply transfinite Dirichlet box?
I am not sure what you mean by this.
However, forcing might be better thought of as comparable to Euclid's proof that there is no greatest prime.
> 2) are there any results due transfinite cardinals, not of transfinite > cardinals?
The Borel hierarchy is defined in terms of the first uncountable ordinal. Hence, results in descriptive set theory that depend on that definition may count.
I do not have enough knowledge of that branch of study to comment further.
> 3) is not an irregular model of ZF non-well-founded?
What is your definition of irregular?
> 4) does not a model of ZF contain itself?
There are relativizations of models. So, one question in set theory is whether
where HOD are the hereditarily ordinal-defined sets and HOD^HOD is HOD relativized within itself.
In this sense, models may have representations within themselves. But, once again, expertise is lacking here.
> 5) is ZF not a model of itself?
ZF is an axiomatization. The question is not well-construed.